Tuning Rashba spin-orbit coupling in homogeneous semiconductor nanowires

We use $\mathbf{k}·\mathbf{p}$ theory to estimate the Rashba spin-orbit coupling (SOC) in large semiconductor nanowires. We specifically investigate GaAs- and InSb-based devices with different gate configurations to control symmetry and localization of the electron charge density. We explore gate-controlled SOC for wires of different size and doping, and we show that in high carrier density SOC has a nonlinear electric field susceptibility, due to large reshaping of the quantum states. We analyze recent experiments with InSb nanowires in light of our calculations. Good agreement is found with the SOC coefficients reported in Phys. Rev. B 91, 201413(R) (2015), but not with the much larger values reported in Nat. Commun. 8, 478 (2017). We discuss possible origins of this discrepancy.

Figure 1

Schematic illustration of a NW with a bottom gate. A typical electron gas distribution is shown inside (yellow). A schematic of the semiconductor band structure used within the Kane model is shown. Symbols indicate conduction (c), heavy-hole (hh), light-hole (lh), and split-off (so) bands with corresponding group-theoretical classification of zone center states.

Figure 2

SOC coefficients ${\alpha }_x^{11}$ and ${\alpha }_y^{11}$ as a function of the gate voltage $V_g$ for three different gate configurations, as shown in the top-left insets. (a) bottom, (b) left-bottom, and (c) left-bottom-right gates. In each panel, insets show the self-consistent electron density at gate voltages $V_g=-0.4,0,0.4$ V.

Figure 3

Cross-sections of the self-consistent potential $V$ (red line, left axis), electron density distribution $n_e$ (blue line, right axis), and $|{\psi }_1{(x=0,y)|}^2$ (green line) along the diameter in the $y$ direction for the gate voltages (a) $V_g=-0.4$, (b) 0, (c) 0.4 V. The two components of the self-consistent potential $V$, namely, the gate voltage $V_{\mathrm{gate}}$ and the electron-electron interaction $V_H$, are shown in red dashed and dotted lines, respectively. Arrows attached to the potential curves denote the electric field direction. Calculations correspond to the bottom-gate configuration of Fig. 2.

Figure 4

${\alpha }_x^{11}\left(V_g\right)$ calculated under the assumption of (a) a constant chemical potential $\mu$, and (b) a constant electron density $n_e$. Inset in panel (a) shows the comparison between the SOC coefficients of the three lowest subbands, calculated with constant $\mu =0.85$ eV. (c) Same as panel (b) but zooming around symmetry point $V_g=0$. (d) The SOC electric susceptibility ${\alpha }_x^{11}$ at $V_g=0$ as a function of the electron concentration $n_e$.

Figure 5

Left column: Maps of electron density $n_e(x,y)$ (left) and envelope function ${\psi }_1(x,y)$ (right) for gate voltage (a) $V_g=-0.01$, (b) 0, and (c) 0.01 V. Right column: profile of self-consistent potential $V(x,y)$ (red) and the envelope function ${\psi }_1(x,y)$ (green) along the facet-facet (upper) and edge-edge (lower) directions, as illustrated by the dashed lines in the top-left hexagon. Calculations performed with $n_e={10}^7 {\mathrm{cm}}^{-1}$.

Figure 6

Same as Fig. 5 for gate voltage (a) $V_g=-0.01$, (b) 0, and (c) 0.01 V. Calculations performed with $n_e=3\times {10}^9 {\mathrm{cm}}^{-1}$.

Figure 7

Absolute value of the intersubband SOC couplings ${\alpha }_{x\left(y\right)}^{12}$ and ${\alpha }_{x\left(y\right)}^{13}$ as a function of the gate voltage $V_g$. Inset shows $E_n\left(V_g\right)$ for the four lowest electronic states. Results for $n_e={10}^7 {\mathrm{cm}}^{-3}$.

Figure 8

Envelope functions of the three lowest electronic states together with the self-consistent potential $V(x,y)$ for $V_g=-0.2$ and 0.2 V.

Figure 9

(a) The intrasubband ${\alpha }_x^{11}$ and (b) intersubband ${\alpha }_y^{12}$ couplings as a function of the NW width $W$, for different electron concentrations, as indicated, at $V_g=0.4$ V.

Figure 10

${\alpha }_x^{11}$ vs $V_g$ calculated by the assumption of (a) the constant chemical potential and (b) the constant electron density. Results for $W=87$ nm and the “ideal” bottom gate configuration.

Figure 11

(a) Schematic illustration of the simulated device. Two gates are connected to the wire through dielectric layers, as indicated, held at voltages $V_{bg}$ and $V_{tg}$. (b) ${\alpha }_x^{11}$ as a function of the bottom gate voltage $V_{bg}$, with $V_{tg}=0$. (c) ${\alpha }_x^{11}$ as a function of the top gate voltage $V_{tg}$, with $V_{bg}=0$. Insets in (b) and (c) show the electron concentration $n_e(x,y)$ (top) and square of the envelope functions of the ground state $|{\psi }_1{(x,y)|}^2$ (bottom) at selected gate voltages indicated by arrows. In (c), the range of measured SOC coefficients [34] is marked by the gray area. (d) Map of ${\alpha }_x^{11}$ as a function of both the gate voltages $V_{tg}$ and $V_{bg}$. Results for $\mu =0.35$ eV.

Figure 12

(a) SOC couplings ${\alpha }_x^{nn}$ as a function of the top gate voltage $V_{tg}$ calculated for the four lowest subbands ($V_{bg}=0$). The corresponding ${\left|{\psi }_n\right|}^2$ are shown in the inset for $V_{tg}=0.6$ V (vertical dashed line), together with the $y$ component of the electric field. (b) Intersubband SOC couplings ${\alpha }_y^{12}$ and ${\alpha }_x^{13}$ as a function of the top gate voltage $V_{tg}$.

Figure 13

(a) Schematic illustration of the experimental setup in Ref. [15]. The conductance of the NW is controlled by the bottom gate $V_{bg}$ attached to the wire by the 20-nm-thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ layer. (b) ${\alpha }_x^{11}$ as a function of the bottom gate $V_{bg}$ for the NW without the sulfur layer (black curve), with the sulfur layer (red curve), and with the dopant concentration included (green curve).